Related papers: A Matrix Integral Solution to [P,Q]=P and Matrix L…

This paper presents a new formula for the q-shift operator, building on the techniques by Liu and Sears. This formula provides fresh proof of the Carlitz formula and extends it naturally. As applications, we derive an equivalent form of the…

Matrix integrals used in random matrix theory for the study of eigenvalues of matrix ensembles have been shown to provide $ \tau $-functions for several hierarchies of integrable equations. In this paper, we construct the matrix integral…

We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to $\mathbb{Z}_2^d$. Noteworthy, we admit negative values of the…

We describe a new operator space structure on $L_p$ when $p$ is an even integer and compare it with the one introduced in our previous work using complex interpolation. For the new structure, the Khintchine inequalities and Burkholder's…

The matrix code equivalence problem consists, given two matrix spaces $\mathcal{C},\mathcal{D} \subset \mathbb{F}_q^{m\times n}$ of dimension $k$, in finding invertible matrices $P\in\mathrm{GL}_m(\mathbb{F}_q)$ and…

Lattices of polynomial KP and BKP $\tau$-functions labelled by partitions, with the flow variables equated to finite power sums, as well as associated multipair KP and multipoint BKP correlation functions are expressed via generalizations…

One of the features of Baxter's Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to…

We present two new algorithms for the computation of the q-integer linear decomposition of a multivariate polynomial. Such a decomposition is essential for the treatment of q-hypergeometric symbolic summation via creative telescoping and…

The efficient implementation of matrix arithmetic operations underpins the speedups of many quantum algorithms. We develop a suite of methods to perform matrix arithmetics -- with the result encoded in the off-diagonal blocks of a…

We report a new analytical method for solution of a wide class of second-order differential equations with eigenvalues replaced by arbitrary functions. Such classes of problems occur frequently in Quantum Mechanics and Optics. This approach…

In the present paper the authors consider the $\mathcal{P}_{\nu,2}$-transform as a generalization of the Widder potential transform and the Glasser transform. The $\mathcal{P}_{\nu,2}$-transform is obtained as an iteration of the the…

We solve the problem of Fourier transformation for the one-dimensional $q$-deformed Heisenberg algebra. Starting from a matrix representation of this algebra we observe that momentum and position are unbounded operators in the Hilbert…

We deal with the existence of three distinct solutions for a poly-Laplacian system with a parameter on finite graphs and a $(p,q)$-Laplacian system with a parameter on locally finite graphs.The main tool is an abstract critical points…

If $Q$ is a non degenerate quadratic form on ${\bb C}^n$, it is well known that the differential operators $X=Q(x)$, $Y=Q(\partial)$, and $H=E+\frac{n}{2}$, where $E$ is the Euler operator, generate a Lie algebra isomorphic to ${\go…

The graph Laplacian is a fundamental object in the analysis of and optimization on graphs. This operator can be extended to a simplicial complex $K$ and therefore offers a way to perform ``signal processing" on $p$-(co)chains of $K$.…

The present work considers one of the simplest homogeneous spaces of the quantum group SU(1,1), the q-analogue of the unit disc in ${\Bbb C}$. We state without proofs q-analogues of Cauchy-Green formulae, integral representations of…

This paper uses the convolution theorem of the Laplace transform to derive new inverse Laplace transforms for the product of two parabolic cylinder functions in which the arguments may have opposite sign. These transforms are subsequently…

Let p and q be conjugate exponents, with p in [1,2]. It is shown that the Laplace transform acts boundedly between the Lp space with unit weight on the positive real semiaxis and the Lq space weighted by a well-projected measure (a term…

Let $P\in \Bbb Q_p[x,y]$, $s\in \Bbb C$ with sufficiently large real part, and consider the integral operator $ (A_{P,s}f)(y):=\frac{1}{1-p^{-1}}\int_{\Bbb Z_p}|P(x,y)|^sf(x) |dx| $ on $L^2(\Bbb Z_p)$. We show that if $P$ is homogeneous…

In this paper we give expressions for the generalized Drazin inverse of a (2,2,0) operator matrix and a $2\times2$ operator matrix under certain circumstances, which generalizes and unifies several results in the literature.